First slide

Transpose of matrix

Question

The matrix $A = (\begin{array}{cc} p & - q \\ q & p \end{array})$ is orthogonal if and only if

Moderate

A

$p^{2} + q^{2} = 1$
B

$p^{2} = q^{2}$
C

$p^{2} = q^{2} + 1$
D

none of these

Solution

$\begin{array}{l} A A^{'} = I \\ \Rightarrow (\begin{array}{cc} p & - q \\ q & p \end{array}) (\begin{array}{cl} p & q \\ - q & p \end{array}) = (\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}) \\ \Rightarrow (\begin{array}{cl} p^{2} + q^{2} & 0 \\ 0 & p^{2} + q^{2} \end{array}) = (\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}) \\ ∴ p^{2} + q^{2} = 1 \end{array}$

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